Brief review

Faster computers and advances in sensor technologies and simulation and control algorithms are removing the main bottlenecks that limited progress in crystallisation control in the 1970s- 1980s. Model identification, experimental design, and optimal control algorithms are being increasingly applied to crystallisation processes in industry, including pharmaceuticals processes which have been resistant to systematic first-principles approaches. Further advances are expected to lead to even more utilization of these techniques to reduce time-to-market, which is key in the pharmaceutical industry, and to increase productivity, which is important in the bulk chemicals industry.

Crystallisation processes have all the characteristics that make an interesting control problem--partial differential equations, nonlinear dynamics, significant uncertainties, unmeasured state variables, significant disturbances, sensor noise, etc. (Braatz, 2002) 2.1.1 Optimal control

An open loop control problem can be formulated where the seed mass, the mean size of seed crystals, the width of the seed crystal size distribution, and the temperature profile are decision variables (Chung et al., 1999; Miller and Rawlings, 1994). Many objective functions have been studied, including the mean size of product crystals, the ratio of standard deviation to mean size, and the ratio of nucleated crystal mass to seed crystal mass at the end of operation (Eaton and Rawlings, 1990).

The optimal solution for each objective function is calculated using successive quadratic programming. A parametric analysis shows the significant importance of optimization of the seed distribution for a wide range of nucleation and growth kinetics (Chung et al., 1999). Under the presence of disturbances, modelling error, or tracking error, the states of the crystalliser do not follow the optimal path. One way to address this problem is to incorporate robustness into the computation of the optimal path (Ma and Braatz, 2000).

However, the performance of this approach will be limited by the chosen measured variables and the use of open loop control.

Several optimal feedback control algorithms including model predictive control have been proposed for batch processes (Eaton and Rawlings; 1990; Rawlings et al., 1993).

Even more recently, feedback control algorithms are being developed to reduce the sensitivity of the product quality to model uncertainties and disturbances, while being applicable to nonlinear distributed parameter systems (Chiu and Christofides, 2000). One approach, which couples geometric control with bilinear matrix inequalities, allows the direct optimization of robust performance (Togkalidou and Braatz, 2000; VanAntwerp et al., 1997; VanAntwerp et al., 1999). In contrast to most approaches to robust nonlinear control, this approach introduces no conservatism during the controller synthesis procedure. Also, no prior limitations are required regarding the speed of the unmodeled dynamics; instead, engineering intuition is incorporated into weights which bound the unmodeled dynamics, similarly to the linear time invariant case (Morari and Zafiriou, 1989; Skogestad and Postlethwaite, 1996). Application to a crystallisation process demonstrated robustness to a wide range of nonlinear and time-varying perturbation (Togkalidou and Braatz, 2000).

Usually the feedback controller is designed to follow a temperature trajectory that comes from, for example, solving an open loop optimal control problem. It has been conjectured, however, that a lower sensitivity to parameter uncertainties and disturbances may result from using the solution concentration as a function of temperature as the setpoint trajectory instead (Gutwald and Mersmann, 1990). Such a formulation, which includes time only as an implicit variable in the setpoint trajectory, can be used in formulating either open loop or closed loop optimal control design procedures. More research is needed to completely resolve whether such implicit-in-time optimal control formulations are superior to the standard formulation.

2.1.2 Linear and non-linear modelling

Control of crystal quality and in particular crystal size distribution (CSD) and crystal purity is of special interest in crystallisation processes. Control of these properties is challenging due to the complexity and non-linearity of the process and because of the lack of reliable on-line instrumentation to measure the above-mentioned crystal qualities.

The physical model of a crystallisation process consists of a set of coupled non-linear integro-differential equations (mass and energy balances) and a partial differential equation (population balance). This model has to be solved numerically with acceptable speed and accuracy for effective model-based control. Various methods have been suggested for this purpose (Rawlings et al., 1993).

An alternative to the physical modelling is process identification techniques in which the input-output sequences are used to develop linear auto-regressive exogenous (ARX) models for the crystallisation process. Previous research (Jager et al., 1992) recommends that in order to improve the crystallisers control, as many inputs and outputs as possible must be exploited. There have been, however, relatively few studies and experimental implementation of multivariable controllers on continuous crystallisers. Those few studies which have attempted multivariable control of crystallisation processes, have used linear models in spite of the fact that both the steady-state and dynamic behaviour of the process exhibit strong non-linearities.

2.1.3 Model predictive control

For control of a system which is represented by a nonlinear model, two general approaches may be taken. The first approach is to linearise the model and use linear control design techniques. Small signal approximation around the operating condition, exact linearisation using differential geometric theory and variable transformation are among various techniques (Daoutidis and Kravaris, 1991; Kravaris and Soroush, 1990), employed for linearization of linear models. The second approach involves non-linear control design techniques. Non-non-linear model-based control design, and more specifically non-linear model predictive control (NMPC), and the differential geometric approach, are common techniques for control of a system which is represented by a nonlinear model. The main advantage of the NMPC approach is its intuitive interpretation and its ability to deal with variables constraints whereas the geometric approach has a solid theoretical foundation. The relation between the two non-linear control techniques has been discussed by Soroush and Kravaris (1993). Model predictive control has received wide acceptance in industry (Qin and Badgwell, 2003). The predecessors of the non-linear MPC algorithm, namely the dynamic matrix control

(DMC) (Culter and Ramaker, 1980) and model algorithmic control (MAC) (Mehra and Rouhani, 1980) use the non-parametric impulse and step responses of the system as the process model. A quadratic objective function is minimized which for an unconstrained minimization, results in a closed-form solution. Proposed non-linear MPC techniques range from a simple extension of DMC, based on successive linearization of the non-linear model (Garcia, 1984; Gattu and Zafiriou, 1992), to more elaborate and computationally intensive techniques involving discretisation of the model followed by solution via non-linear programming (Bequette, 1991; Biegler and Rawlings, 1991).

Application of neural networks in control has been a fast growing research activity in the past few decades and may have a promising future. Neural networks have been used as a feedforward model, as inverse model in supervisory control, and as an optimizer. A good survey study on the application of neural networks in control is given in Hunt et al.

(1992).

In non-linear model predictive control, a solution of the non-linear dynamic model is required in each optimization iteration. In quadratic dynamic matrix control (QDMC) (Garcia and Morshedi, 1986) the process models linearised once for the entire trajectory, whilst in nonlinear quadratic matrix control (NLQDMC), the process model is linearised iteratively in each control interval (Lee and Recker, 1994). Exact linearisation using Lie derivatives has been employed in Soroush and Kravaris (1993) in which the exact inverse of the process model is obtained. However, for non-minimum phase processes such as a crystallisation process, exact linearisation results in an unstable inverse. An alternative to the sequential solution is to use a simultaneous solution of the process model and the optimisation algorithm. In this approach, model equations are transformed to algebraic equations, which reduce an infinite-dimensional optimization problem (optimal control) to a finite-dimensional problem (non-linear programming) (Morshedi, 1986). The ability of neural network-based model predictive control has been discussed by many workers including Dayal et al. (1994); Hunt and Sarbaro (1991); Nahas et al.

(1992); Psichogios and Ungar (1991); Takahashi (1993). Little work has been reported on the multivariable control of crystallisation processes (Jager et al., 1992). This is primarily due to the lack of reliable techniques for the on-line measurement of crystal qualities of interest, namely; crystal size distribution (CSD), crystal morphology, and

crystal purity. On the other hand, dynamic modelling of the crystallisation processes is still rudimentary, especially in predicting the nucleation rate, growth rate, agglomeration rate, and other number-affecting phenomena (Miller and Rawlings, 1994). Attempts have been made by Eek (1995) to linearise the non-linear model of a crystalliser followed by model reduction and linear model predictive control. In de Wolf et al. (1989) and Jager et al. (1992) an ARX model structure was used to identify a multi-input-single-output (MISO) model for a continuous crystalliser which was then used to derive the state-space formulation of the process. In Myerson et al. (1987) a multivariable optimal control of a mixed-suspension mixed-product removal crystalliser was studied. The controller was based on the linearised moments equations and mass and energy balance equations. An extended Kalman filter was used for state estimation.

In the paper of Rohani (1999/2) non-linear modelling and control of a continuous crystallisation process uses the combined concepts of neural networks (for modelling) and non-linear model predictive control (for control). In this outstanding study (Rohani, 1999/2) the linear ARX model (for comparison) and the non-linear feedforward neural network with parallel-parallel connection were used as process models. The control is achieved using the model predictive control strategy. This is an extension of the work by Eek (1995) in which he considered only the linear quadratic model predictive control of a crystallisation process. The use of neural network in non-linear model predictive control approach alleviates the main disadvantage of using the non-linear physical model which demands high computational load in each optimization step.

To summarize, the study of the crystallisation processes has so far mostly relied on linearised models and therefore application of linear control theory. (Such as this dissertation.) This is in spite of the fact that both steady-state and dynamic behaviours of the process exhibit strong non-linearities.